Tuning of the flat band and its impact on superconductivity in Mo5Si3−xPx

The superconductivity in systems containing dispersionless (flat) bands is seemingly paradoxical, as traditional Bardeen-Cooper-Schrieffer theory requires an infinite enhancement of the carrier masses. However, the combination of flat and steep (dispersive) bands within the multiple band scenario might boost superconducting responses, potentially explaining high-temperature superconductivity in cuprates and metal hydrides. Here, we report on the magnetic penetration depths, the upper critical field, and the specific heat measurements, together with the first-principles calculations for the Mo5Si3−xPx superconducting family. The band structure features a flat band that gradually approaches the Fermi level as a function of phosphorus doping x, reaching the Fermi level at x ≃ 1.3. This leads to an abrupt change in nearly all superconducting quantities. The superfluid density data placed on the ’Uemura plot‘ results in two separated branches, thus indicating that the emergence of a flat band enhances correlations between conducting electrons.

A successful phosphorus doping into the Mo 5 Si 3 matrix is confirmed by the corresponding changes of a− and c−lattice constants, as shown in Fig. 1 (b).The phosphorus content in synthesized Mo 5 Si 3−x P x samples was determined by comparing the lattice parameters a and c with those reported in Ref. 1. Figure 1 (c) implies that the measured P concentrations agree well with the nominal ones.Consequently, the nominal phosphorus content x was used for representing the data in the main text.

III. SPECIFIC HEAT DATA
The temperature dependencies of the specific heat (C p ) of Mo 5 Si 3−x P x are presented in Fig. 3 (a).Note that the C p (T ) data were corrected by subtracting the minor contributions of Mo 3 P.The temperature evolution of C p in the normal state (i.e., for T c < T 16 K) was analyzed within the framework of the Debye model: where the linear T term represents the electronic specific heat contribution γ, while T 3 and T 5 terms account for the harmonic and anharmonic phonon contributions, respectively.Fits of the Debye model to C p (T ) data are presented in Fig. 3 (a) by dotted lines.The Debye temperature Θ D was further obtained by using the 'harmonic' term β as: Here N is the number of atoms per formula unit, and R is the ideal gas constant.The dependence of Θ D on x is shown in Fig. 3  The absence of a competing ordered state(s), which might be responsible for reducing the number of carriers accessible for the superconducting condensate, was checked by comparing the Density of States (DOS) as obtained from the first principle calculations [Fig.4 (a)] with DOS at the Fermi level [N (E F )] determined from the above reported quantities (γ and λ ep ) via: An agreement between the 'theoretical' and 'experimental' dependencies of N (E F ) on x, as presented in Fig. 4 (b), confirms the absence of competing states formed above the superconducting transition temperature T c .

V. ANALYSIS OF λ −2 (T ) DEPENDENCIES
The individual temperature dependencies of the inverse squared magnetic penetration depth λ −2 of Mo 5 Si 3−x P x , as obtained in TF-µSR experiments, are presented in Fig. 5.The parameters obtained from the fit of the s−wave gap model (Eq. 3 in the main text) to λ −2 (T ) data, namely the superconducting transition temperature T c , the zero temperature value of the inverse squared magnetic penetration depth λ −2 (0), and the superconducting energy gap ∆(0), are shown at the corresponding panels.) ) ) ) ) x = 1 .4 ) x = 1 .5 ) x = 1 .6 The zero-field µSR measurements were performed in order to reveal a possible breaking of the time-reversal symmetry, which may indicate an unconventional superconducting state of Mo 5 Si 3−x P x system.The ZF-µSR time spectra for x = 1.2 and x = 1.4 samples, collected above the superconducting transition temperature T c and at T 1.5 K, are presented in Fig. 6.Neither coherent oscillations nor fast decays could be seen, thus excluding any type of magnetic order or fluctuations.
The zero-field µSR spectra were fitted using the Gaussian Kubo-Toyabe (GKT) relaxation function, 3,4 describing the nuclear moment response, multiplied by an additional exponential term: Here, A 0 is the initial asymmetry, σ GKT is the GKT relaxation rate, and Λ is the exponential relaxation rate.In the above equation, σ GKT accounts for the nuclear moment contribution, which is assumed to be static within the µSR time window and independent on temperature.

x
FIG. 1: (a) Powder x-ray diffraction patterns of Mo5Si3−xPx (0 ≤ x ≤ 1.6).The peaks of the Mo3P impurity phase are marked by asterisks.The inset shows the position of (141) peak of Mo5Si3−xPx at various doping levels.(b) The dependence of the lattice parameters a and c on the nominal phosphorus content x.(c) The measured phosphorus content versus its nominal value.The dashed line represents the case where the measured values are equal to the nominal ones.

Figure 2
Figure2shows the resistivity curves normalized to the values at T = 16 K [R(T )/R(16 K)] measured in magnetic fields ranging from 0.0 to 9.0 T. The superconducting transition temperature at the applied field µ 0 H ap [T c (µ 0 H ap )] was determined as a cross point of R(T, H ap )(T ) curve with R(T )/R(16 K) = 0.5 line, see the top left panel of Fig.2.

FIG. 2 :
FIG. 2: Temperature dependencies of resistivity of Mo5Si3−xPx under magnetic fields ranging from 0.0 to 9.0 T. The superconducting transition temperature Tc is defined from the midpoint of R(T, Hap) curves [i.e., as the value where R(T )/R(16 K) = 0.5, see the top left panel].

FIG. 3 :
FIG. 3: (a) Temperature dependencies of the specific heat Cp of Mo5Si3−xPx under zero magnetic field.The dash lines correspond to the fit of the Debye model (Eq. 1) to the Cp(T ) data.The kinks at T 2 30 K arise from the Mo3P impurity.(b) Dependence of the Debye temperature (ΘD) on the phosphorus content x.(c) Dependence of the electron-phonon coupling constant (λep) on x.

FIG. 6 :
FIG.5: Temperature dependencies of the inverse squared magnetic penetration depth λ −2 of Mo5Si3−xPx as obtained in TF-µSR experiments.The solid lines are fits of Eq. 3 from the main text to λ −2 (T ) data.The fit parameters, namely the superconducting transition temperature Tc, the zero temperature value of the inverse squared magnetic penetration depth λ −2 (0), and the superconducting energy gap ∆(0), are presented at each corresponding panel.

Figure 7
Figure7shows the temperature dependence of the exponential relaxation rate Λ of Mo 5 Si 3−x P x samples.The absence of an additional µSR relaxation below T c excludes a possible time-reversal symmetry breaking in the superconducting state of Mo 5 Si 3−x P x .